By the results in [1]}, [2]}, each \(\varphi \in {\mathcal {E}}(M, \overline{\omega }_0)\)  can be weakly approximated (as a current) by a sequence of smooth functions \((\varphi _j)_j \in {\mathcal {H}}(M, \overline{\omega }_0)\)  and for any such sequence \(\lim _{j\rightarrow \infty } \overline{\omega }_{\varphi _j}^{[m]} = \overline{\omega }_{\varphi }^{[m]}\)  in the sense of measures. We can take the latter as a definition of \(\overline{\omega }_{\varphi }^{[m]}\)  when \(\varphi \in {\mathcal {E}}(M, \overline{\omega }_0)\) . Following [1]}, one further introduces the sub-space of elements with full Monge-Ampère mass and finite energy
\({\mathcal {E}}^1(M, \overline{\omega }_0) = \Big \lbrace  \varphi \in {\mathcal {E}}(M, \overline{\omega }_0) \, \Big | \, \int _M |\varphi | \, \overline{\omega }_{\varphi }^{[m]} < \infty \Big \rbrace .\) 
